Efficient design of digital filters for 2-D and 3-D depth migration

Abstract

Two-and three-dimensional depth migration can be performed using one-and twodimensional extrapolation digital lters, respectively. The depth extrapolation is done, one frequency at a time, by convolving the seismic wave eld with a complex-valued, frequencyand velocity-dependent, digital lter. This process requires the design of a complete set of extrapolation lters, one lter for each possible frequency-velocity pair. Instead of independently designing the frequency-and velocity-dependent lters, an e cient procedure is introduced for designing a complete set of one-and two-dimensional extrapolation lters using transformations. The problem of designing a desired set of migration lters is thus reduced to the design of a single one-dimensional lter which is then mapped to produce all the desired one-or two-dimensional migration lters. The new design procedure has the additional advantage that both the one-and two-dimensional migration lters can be realized e ciently and need not have their coe cients precomputed or tabulated.

The design of visco‐acoustic weighted L1‐error frequency–space wavefield extrapolators

“…Because of this even symmetry, the extrapolation coefficients are complex valued. Similar to the acoustic case, the ideal visco-acoustic wavenumber response in equation (1) can, hence, be approximated by the design response as (Hale 1991;Karam and McClellan 1997;Yilmaz 2001;Mousa 2014):…”

“…Depth migration is considered to be a physically more consistent and accurate domain to compensate and improve the resolution of the data at higher depths, as reported by Mittet, Sollie and Hokstad (1995) and Mittet (2007). In most of the previous research work, researchers used different methods to design non-compensating acoustic medium wavefield extrapolators in the f -x domain (Holberg 1988;Hale 1991;Soubaras 1996;Karam and McClellan 1997;Thorbecke 1997;Thorbecke et al 2004;Mousa et al 2009;Mousa 2014;Naseer and Mousa 2015), whereas in Mittet (2007), visco-acoustic extrapolators are designed using the L 4 -norm. In this paper, f -x extrapolators are designed to compensate for attenuation using the law of absorption for a linear visco-acoustic medium (Mittet et al 1995).…”